measures of dispersions

MEASURES OF DISPERSIONS

MEASURES OF DISPERSIONS

• A quantity that measures the variability among the data, or how the data one dispersed about the average, known as Measures of dispersion, scatter, or variations.

• To know the average variation of different values from the average of a series

• To know the range of values • To compare between two or more series

expressed in different units • To know whether the Central Tendency

truly represent the series or not

2. Common Measures of Dispersion

• The main measures of dispersion1. Range

2. Mean deviation or the average deviation

3. The variance & the standard deviation

5

The Range

• The range is defined as the difference between the largest score in the set of data and the smallest score in the set of data, XL - XS

• What is the range of the following data:4 8 1 6 6 2 9 3 6 9

• The largest score (XL) is 9; the smallest score (XS) is 1; the range is XL - XS = 9 - 1 = 8

Ahad Kabir

1. RANGE• Example:

1. Find the range in the following data.

31,26,15,43,19,10,12,37

Range = xm – xo 33 = 43 – 10

2. Find the range in the following F.D. (Ungrouped)

5 = 8 – 3

Range 5 = 8 – 3

3. Find the range in the following data.

Range = 60 – 10 = 50

X 3 4 5 6 7 8

f 5 8 12 10 4 2

X 10 - 20 20 - 30 30- 40 40 – 50 50 - 60

f 5 8 12 10 4

MEAN (OR AVERAGE) DEVIATION

• It is defined as the “Arithmetic mean of the absolute deviation measured either from the mean or median.

• or for ungroup.

• or for grouped.

n

xxDM

∑ −=..

N

xxf∑ −=

N

medianx∑ −

N

medianxf∑ −=

Example : 

Sazzad took five exams in a class and had scores of 92, 75, 95, 90, and 98. Find the mean 

deviation for his test scores.  

               We can say that on the average, Saddam’s test scores deviated by 6 points from the mean.

Atik hasan

MEAN (OR AVERAGE) DEVIATION

• Exp: Calculate mean deviation from the FD (Grouped Data).

MD (x) =    33.6 / 20 = 1.68 

M.D = 23.72 / 14 = 1.69

X f Class Mark ( x )

f.x I x – 6.57 I f I x – 6.57 I

2 – 4 2 3 6 3.57 7.14

4 - 6 3 5 15 1.57 4.71

6 – 8 6 7 42 0.43 2.58

8 – 10 2 9 18 2.43 4.86

10 – 12 1 11 11 4.43 4.43

Total Σf =14 Σ f.x =92 Σ f I x – 6.57 I = 23.72

=92/14=6.57ẋ

• It is an absolute measure.

• It’s relative measure is coefficient of M.D.

• Coefficient of M.D. = 

• It is based on all the observed values.

MEAN (OR AVERAGE) DEVIATION

median

DMor

mean

DM ....

THE VARIANCE ANDSTANDARD DEVIATION

• It is defined as “The mean of the squares of deviations of all the observation from their mean.” It’s square root is called “standard deviation”. 

• Usually it is denoted by    (for population of statistics)  S2 (for sample) 

•   =    for ungrouped

2σ

2σn

xx∑ − 2)(

Bakhtiare Hossain

•   =  for grouped

• It is an absolute measure;

• It is relative measure is coefficient of variation.

•  

• Shortcut method

N

xxf∑ − 2)(2σ

100. ×=µσ

VC 100..

.. ×=x

DSVC

222

−= ∑∑

N

x

N

xσ

222 .

−= ∑∑

N

fx

N

xfσ

THE VARIANCE ANDSTANDARD DEVIATION

VARIANCE AND STANDARD DEVIATION

• Example:

1. Calculate  Variance and SD from the FD (Ungrouped Data).

Using Short cut method

var = (564 / 20) -    (98 /  20) ^ 2  =   28.2 – 24.01 = 4.09

Sd = √ σ^2 = √ 4.09 = 2.02

X f f.x X^2 f.x^2

2 3 6 4 12

4 9 36 16 144

6 5 30 36 180

8 2 16 64 128

10 1 10 100 100

Total Σf =20 Σf.x = 98 Σ f.x^2=564

222 .

−= ∑∑

N

fx

N

xfσ

VARIANCE AND STANDARD DEVIATION

• Exp: Calculate Variance and Standard deviation from the FD (Grouped Data).

Using Short cut method:

var = (670 /14) - (92 / 14) ^ 2 = 47.85 – 43.18 = 4.67

Sd = √ σ^2 = √ 4.67 = 2.16

X f Class Mark ( x )

f.x x^2 f.x^2

2 – 4 2 3 6 9 18

4 - 6 3 5 15 25 75

6 – 8 6 7 42 49 294

8 – 10 2 9 18 81 162

10 – 12 1 11 11 121 121

Total Σf =14 Σ f.x =92 Σ f.x^2 =670

222 .

−= ∑∑

N

fx

N

xfσ

Imran hossain

Relative Measures of Relative Measures of DispersionDispersion

Coefficient of Range Coefficient of Quartile Deviation Coefficient of Mean Deviation Coefficient of Variation (CV)

Relative Measures of Variation Relative Measures of Variation

Largest Smallest

Largest Smallest

Coefficient of RangeX X

X X

−=

+

3 1

3 1

Coefficient of Quartile DeviationQ Q

Q Q

−=

+

Coefficient of Mean DeviationMD

Mean=

Coefficient of Variation (CV)Coefficient of Variation (CV)

Can be used to compare two or more sets of data measured in different units or same units but different

average size.

100%X

SCV ⋅

=

Use of Coefficient of VariationUse of Coefficient of VariationStock A:

Average price last year = $50Standard deviation = $5

Stock B:Average price last year = $100Standard deviation = $5 but stock B is

less variable relative to its price

10%100%$50

$5100%

X

SCVA =⋅=⋅

=

5%100%$100

$5100%

X

SCVB =⋅=⋅

=

Both stocks have the same standard deviation

Arjun Baidya

SkewnessA fundamental task in many statistical analyses is to

characterize the location and variability of a data set (Measures of central tendency vs. measures of dispersion)

Both measures tell us nothing about the shape of the distribution

It is possible to have frequency distributions which differ widely in their nature and composition and yet may have same central tendency and dispersion.

Therefore, a further characterization of the data includes skewness

Positive & Negative SkewPositive skewness

There are more observations below the mean than above it

When the mean is greater than the medianNegative skewness

There are a small number of low observations and a large number of high ones

When the median is greater than the mean

Measures of SkewSkew is a measure of symmetry in the distribution

of scores

Positive Skew

Negative Skew

Normal (skew = 0)

Measures of Skew

Robiul Sarkar

The Kurtosis is the degree of peakedness or flatness of a unimodal (single humped) distribution,

• When the values of a variable are highly concentrated around the mode, the peak of the curve becomes relatively high; the curve is Leptokurtic.

• When the values of a variable have low concentration around the mode, the peak of the curve becomes relatively flat;curve is Platykurtic.

• A curve, which is neither very peaked nor very flat-toped, it is taken as a basis for comparison, is called Mesokurtic/Normal.

Measures of Kurtosis

Measures of Kurtosis

Measures of Kurtosis

1. If Coefficient of Kurtosis > 3 ----------------- Leptokurtic.

2. If Coefficient of Kurtosis = 3 ----------------- Mesokurtic.

3. If Coefficient of Kurtosis < 3 ----------------- is Platykurtic.

( )( )

4

22

n X-XCoefficient of Kurtosis=

X-X

∑∑